1. From the Cardano reading and videos you can see that he can solve the "depressed cubic" (no x² term) in any of its variations. The general cubic can be turned into a depressed cubic, so he could solve all cubics. Verify this using modern notation: a. Start with the general cubic ax³ + bx² + cx+d = 0, make the substitution x = y-, b За and simplify. Gather all the like terms in y. Write each of your coefficients of the y terms (and constant term) as a single fraction (i.e., get a common denominator). Do not be afraid of a little algebra! Naturally, the 12 terms should cancel out. b. This should leave you with a "depressed cubic" which you should write in the form y³ + Cy=D. [Don't forget to divide by a.] Write out what C and D are equal to in terms of a, b, c, d. Do not be afraid of a little algebra!

1. From the Cardano reading and videos you can see that he can solve the "depressed cubic" (no x² term) in any of its variations. The general cubic can be turned into a depressed cubic, so he could solve all cubics. Verify this using modern notation: a. Start with the general cubic ax³ + bx² + cx+d = 0, make the substitution x = y-, b За and simplify. Gather all the like terms in y. Write each of your coefficients of the y terms (and constant term) as a single fraction (i.e., get a common denominator). Do not be afraid of a little algebra! Naturally, the 12 terms should cancel out. b. This should leave you with a "depressed cubic" which you should write in the form y³ + Cy=D. [Don't forget to divide by a.] Write out what C and D are equal to in terms of a, b, c, d. Do not be afraid of a little algebra!

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter4: Polynomials

Section: Chapter Questions

Problem 40CLR

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